Integrals and crossed products over weak Hopf algebras

In this paper we present the general theory of cleft extensions for a cocommutative weak Hopf algebra $H$. For a weak left $H$-module algebra we obtain a bijective correspondence between the isomorphisms classes of $H$-cleft extensions $A_{H}\hookrightarrow A$, where $A_{H}$ is the subalgebra of coinvariants, and the equivalence classes of crossed systems for $H$ over $A_H$. Finally, we establish a bijection between the set of equivalence classes of crossed systems with a fixed weak $H$-module algebra structure and the second cohomology group $H_{\varphi_{Z(A_H)}}^2(H, Z(A_H))$, where $Z(A_H)$ is the center of $A_H$.